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My textbook says $|a| = -a$ for $a \le 0$.

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What does it mean by this? I'm confused and think that the absolute value of a negative number like $|-5|$ would be positive $5$?

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user3765756
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    But $5$ is indeed $-(-5)$. – lulu Sep 01 '18 at 16:40
  • Yea, I figured it was some small detail like that just don't have an intuitive grasp of it. So basically its a way of retaining the sign for the original value and you would think of it like -(-5)? That makes sense, thanks @lulu – user3765756 Sep 01 '18 at 16:42
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    $-a$ does not mean "a number that is negative." It means "the additive inverse (or 'opposite') of $a$". The opposite of a negative number is a positive number. So if $a < 0$ (i.e. $a$ is negative), then $-a > 0$ (i.e. $-a$ is positive). – Xander Henderson Sep 01 '18 at 16:43
  • Ahh ok I think I understand the concept more now. Thanks @XanderHenderson. I'm new to this site so not sure if I need to approve these answers or show my thanks by giving a rating but I appreciate the help. – user3765756 Sep 01 '18 at 16:46
  • The absolute value is the "non-negative magnitude". If $a > 0$ then $|a| = a>0$. But if $a < 0$ then notice that $-a > 0$. So that means that $-a$ is be positive magnitude and $a< 0$ is the negative magnitude. – fleablood Sep 01 '18 at 16:49
  • Yeah, for me the trick is realizing that if $a < 0$ then $-a$ *IS* a positive number (but only if $a < 0$. If $a > 0$ then $-a$ is negative and if $a=0$ then $-a = a = 0$.) – fleablood Sep 01 '18 at 16:54
  • You might think of the absolute value of a number as the distance away from zero (which has helped my intuition). $-5$ is therefore five units way from zero. – Mark Bennet Sep 01 '18 at 16:57

3 Answers3

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According to the definition

  • for $x\ge 0$ we have $|x|=x$

  • for $x< 0$ we have $|x|=-x$

we have

  • $|5|=5$

  • $|-5|=-(-5)=5$

therefore $|x|$ is always non negative.

user
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You are correct $|-5| = 5$. BUT notice if $a = -5$ then $-a = -(-5) = 5$.

Think of it this way. If $a < 0$ then $|a|$ is positive and $a$ is negative. So $|a|$ is the opposite of $a$. So $|a| = -a$ because $-a$ is the opposite of $a$.

In short. An negative sign before a variable does not mean the number is negative. It means the number is the opposite sign of whatever the variable means.

To beat a dead horse: If $a \ge 0$ then $|a| > 0$. But if $a < 0$ then $|a|$ would be "whatever it takes to make $a$ positive" which is, as "a negative of a negative is a positive" must be the other direction and that is $-a$. And we know $-a > 0$.

.... or in yet another way....

The absolute value is the "non-negative magnitude". If $a > 0$ then $|a| = a>0$. But if $a < 0$ then notice that $-a > 0$. So that means that $-a$ IS the positive magnitude and because $a< 0$, then $a$ is the negative magnitude.

So $|a| = -a$.

Which is intuitively the opposite of what we think. Your confusion is understandable and common. But logically unnecessary.

fleablood
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For $a \le 0$, we indeed have $|a| = (-1)(a)$. You want to try $a=-5$, so we have

$$|\underbrace{-5}_{a}|=|a|= (-1) \ \underbrace{(a)}_{a=-5}=-(-5)=5$$

Don't forget, $a$ isn't always positive.

  • For positive numbers, the absolute value of $a$ is given by multiplying $1$ with $a$. Eg $|5|=(1)(5)=5$. So, for positive numbers, the absolute value of $a$ is itself.

  • For negative numbers, the absolute value of $a$ is given by multiplying by $-1$ with $a$. Eg $|-5|=(-1)(-5)=5$. So, for negative numbers, the absolute value of $a$ is $-a$.

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